Problem: Simplify the following expression and state the condition under which the simplification is valid. $q = \dfrac{-5t^3 - 10t^2 + 175t}{-6t^3 - 96t^2 - 378t}$
Explanation: First factor out the greatest common factors in the numerator and in the denominator. $ q = \dfrac {-5t(t^2 + 2t - 35)} {-6t(t^2 + 16t + 63)} $ $ q = \dfrac{5t}{6t} \cdot \dfrac{t^2 + 2t - 35}{t^2 + 16t + 63} $ Simplify: $ q = \dfrac{5}{6} \cdot \dfrac{t^2 + 2t - 35}{t^2 + 16t + 63}$ Since we are dividing by $t$ , we must remember that $t \neq 0$ Next factor the numerator and denominator. $ q = \dfrac{5}{6} \cdot \dfrac{(t + 7)(t - 5)}{(t + 7)(t + 9)}$ Assuming $t \neq -7$ , we can cancel the $t + 7$ $ q = \dfrac{5}{6} \cdot \dfrac{t - 5}{t + 9}$ Therefore: $ q = \dfrac{ 5(t - 5)}{ 6(t + 9)}$, $t \neq -7$, $t \neq 0$